166
MORE ABOUT NUMBER-NAMES
usable letters
BILLION TW(
have not foune
sible longest
name using al
756, since F,
It is clear
EDWARD R. WOLPOW
Brookline, Mas s achusetts
This article continues my investigations of the logological proper­
tie s of numbe r - name s begun in the February 1980 Word Ways with
'1 Alphabetizing the Integers".
Following the practice of Webster' s
Third New Inte rnational Dictionary ( see the table on page 1549) , we
consider number-names up to, but not including, one thousand vigin­
tillion. Furthermore, we disregard all 1 ands I and specify that no
powe r of ten can stand alone without a modifier: e. g. , ONE MILLION,
but not MILLION. All calculations reported below have been made
with a hand calculator, so it may be worth verifying and extending these
results with the aid of a high-speed digital computer.
It appears quite certain that there is only one number-name written
entirely with letters from one-half of the alphabet: TWO. There are
only twelve number-names in which vowels (aeiouy) and consonants
alternate: ONE, FIVE, SIX, SEVEN, NINE, TEN, ELEVEN, NINETY,
NINETY - FIVE , NINETY-SIX, NINETY-SEVEN and NINETY-NINE.
The longest of these is NINETY-SEVEN. When considering number­
names beginning and ending with the same letter, the problem is more
complicated. Clearly the smallest and shortest is NINETEEN. Of the
many others, the alphabetically first is EIGHT BILLION EIGHTEEN
MILLION EIGHTEEN THOUSAND EIGHT HUNDRED EIGHTY - FIVE,
and the last is TWO VIGINTILLION TWO UNDECILLION TWO TRIL­
LION TWO THOUSAND TWO HUNDRED TWENTY-EIGHT.
When written out, the longest number-name has 758 letters, and
there are 3 44 or about 10 21 (one sextillion) such numbers. These num­
be r s are gener ated by repeating the triplet x7y 22 time s, whe re x and y
take any of the values 3, 7 or 8. Of all these numbers, the first in al­
phabetical order is the number beginning EIGHT HUNDRED SEVENTY­
EIGHT VIGINTILLION and repeating 21 more 87 8s. The last in alpha­
betical order repeats 373 twenty-two times. If the triad 777 is repeated
22 times, the unique numbe r- na me with the most syllable s, 272, is
produced.
As can be seen from the table of numbers in NI3, 23 or the 26 letters
of the alphabet are used in number-names, lacking only the J, K and Z.
Conside r that the K sound is pre sent in OCTILLION, leaving J and Z.
Interesting, then, that the two non-precise dictionary- sanctioned (NI3)
words for large numbers use just these two letters: JILLION and ZIL­
LION. An exa mple of a 5 6-lette r number - name that uses all the 23
the numbers 1
hundred ( 1) ,
is neglected s
can be writter
possible 10ngE
lette r is SIXT
LION SEVEN~
SEVEN. The
alphabetical 0
QUINTILLIO]\
with eleven UT
eleven units f(
DECILLION S
FORTY-FIVE
Because n
to find sets of
(THREE HUN
easy to find s'
name s in two
ber-names th,
FOUR. What
numbe r- name
233 of Langua
an earlier ref
When an 0
nu.mber-name
c ente r is ONE
fo r 1 through
comes ONE H
be r for 1 thro
1092 and 1 thr
Some of tf
name s are ea
cally first (rE
be ticall y la s t
LION SIXTY ~
the last ordin
TILLION SIX'
There apF
167
usable letters is FIVE SEPTENDECILLION SIX QUADRILLION EIGHT
BILLION TWO MILLION. This 56-letter solution is not uniqu~4 but I
have not found a shorter example. Remarkably, none of the 3
pos­
sible longest number-names include all 23 letters. The longest number­
name using all 23 letters must be two letters short of the maximum, or
756, since F and W cannot be included in any 758-letter solution.
::lgical properWays with
rWebsterls
e 1549) , we
JUsand vigin­
ify that no
~NE MILLION,
been lYlade
extending the se
'-name written
The re are
consonants
'EN, NINETY,
TY -NINE.
ing number­
blem is more
~EEN. Of the
EIGHTEEN
[TY - FIVE,
TWO TRIL­
and
>. These num­
where x and y
le fi r s t in aID SEVENTY­
ast in alpha­
77 is repeated
>,272, is
~tters,
r the 26 letters
e J, K and Z.
ng J and Z.
:tioned (N13)
:)N and ZIL­
all the 23
It is clear that our number-names are built out of 49 linguistic units:
the numbers 1 through 19 ( 19) , the decade numbers frolYl 20 to 90 (8) ,
hundred (1) , thousand (1) and the 20 -illion names. (Centillion, here,
is neglected since it is not continuous with the others.) Many numbers
can be written to include these 49 units just once each. One of the 4096
possible longest number-names using only units beginning with the same
letter is SIXTY-SEVEN SEPTENDECILLION SEVENTY-SIX SEXDECIL­
LION SEVENTY-SEVEN SEPTILLION SIXTY-SIX SEXTILLION SIXTY­
SEVEN. The longest number-name I have found with linguistic units in
alphabetical order is EIGHT HUNDRED T'rINE NOVEMDECILLION ONE
QUINTILLION SEVENTY-SIX THOUSAND TWENTY-TWO (non-unique) ,
with eleven units. Similarly, I could find no examples of more than
eleven units for reverse alphabetical order: TWENTY-THREE SEX­
DECILLION SEVENTY-SEVEN QUADRILLION NINETY - NINE MILLION
FORTY - FIVE (non-unique) .
Because number-names consist of these linguistic units, it is easy
to find sets of letters which form two or more different number-names
(THREE HUNDRED r;'OUR, FOUR HUNDRED THREE). Further, it is
easy to find sets of letters which can be rearranged to forlYl two number­
names in two different ways, with the nu.merical sum of these two num­
ber-names the same: e. g., FOURTEEN + SEVEN = SEVENTEEN +
FOUR. What is surprising is that this can be done for the irregular
number-nameS-ONE + TWELVE = TWO + ELEVEN, a fact noted on page
233 of Language on Vacation ( Sc r ibne r' s, 1965). Can reade r s supply
an earlier reference for this oddity?
When an odd numbe r of numbe r - name s is alphabetized, a single
numbe r - name is in the cente r. In the group ONE through NINE, the
center is ONE. Interestingly, for 1 through 19, it is also ONE, and
for 1 through 99 it is still ONE. For 1 through 999, though, it be­
comes ONE HUNDRED T\TINETY-TWO, which is also the central num­
ber for 1 through 9999. One through 99,999 yields the central nUlYlber
1092 and 1 through 999,999 yields (I think) 117,182.
Some of the problems of reverse alphabetization of the nurrnber­
names are easier to solve than for alphabetical order. The alphabeti­
cally first (reverse) number-name is THREE HUNDRED and the alpha­
betically last is SIX HUNDRED SIXTY VIGINTILLION SIXTY SEPTIL­
LION SIXTY SEXTILLION SIXTY. The first ordinal is SECOND and
the last ordinal is SIX HUNDRED SIXTY VIGINTILLION SIXTY SEP­
TILLION SIXTY SEXTILLION SIXTY - FIRST.
There appears to be no number-name in English that has the same
168
value (when lette r - value s A = 1, B = 2, etc. are summed) as the num­
ber it represents. Some near-misses include 219 (letter-total 218)
and 253 (letter-total 254). At least in the first thousand or so number­
names, the letter E is the most-used letter. It seems that as the - illion
number-names are reached, I might overtake E in letter-frequency. If
so. when doe s this occur?
Let us assume that all of the number-names have been inserted in
an unabr idge d dictionary; which one s appear on the ir cor re ct1y- num­ bered pages? >;'or NI2, there are three matches (822, 1702 and 2748) ,
but for NI3 only two (1576 and 2475). In general, one can imagine in­ serting the first n number-names into any alphabetical list of n pages,
and looking for the number of matches with page-numbers. Probability
theory states that the expected number of such matches is one, but the
actual matches that occur depend upon the detailed structure of the al­ phabetical list. To illustrate this, the editor supplied three lists: NI2
(words in the list spaced proportional to definition-length) , the Air
"5"'orce list of most NI2 words (no definitions) , and the Morris County
1980 telephone directory (surnames spaced proportional to the number
of bearers). Magically shrinking these lists to one to ten pages, one
discovers the matches in the table below.
Pages
1
2
3
6
7
8
9
10
Morris TD
Air Force
NI2
Valid Range
ONE ( .672)
TWO ( .920)
THREE ( .902)
SIX ( .836)
SIX ( .836)
SEVEN ( .820)
ONE ( .565)
TWO ( .898)
THREE ( .867)
ONE ( .570)
TWO'(.920)
THREE ( .881)
SIX ( .782)
SEVEN ( .770)
SEVEN (. 770)
>;'OUR ( .304)
FOUR ( .314)
SIX ( .787)
SEVEN ( .768)
SEVEN ( .768)
FOUR ( .334)
FOUR ( .334)
0-1.0
.50-1.0
.67-1.0
.83-1.0
.71-.86
.75-.88
.67-.78
.33-.44
.30-.40
"For example, in the 500-page Morris TD the number TWO was inserted
on pa~e 460, at the. 920 fractional position in the full list. In a two­
page dictionary, the number TWO matches its page if its fractional pos­
ition lies anywhere in the range of 0.5 to 1.0. Note that none of these
lists creates matches when they are four or five pages in length.
What is the longest number-name ever written out, where the pur­
pose had nothing to do with 10gology? What are the results for these
10gologica1 problems if languages other than English are studied? In
particular, are there solutions in other languages for a number-name
whose 1ette r- sum is equal to the number it represents?
BEYOND
HARRY B. FA
Manhattan Bea
wordst
~able delvers
smelt to fashi<
out an importa
husband and w:
o( the World ' s
1977), in the 1
Prof. Cha des
Ways, distingu
of the hosts of
¥Te
On first tal
"Farsi, Norn, c
failure to find
further' acquaiI
hit upon the ve
authors: nativE
or obsolete Ian
lect rormerly"\.
names from lit
an Iranian dial,
so 'ar as I am
Othe rwise,
odd and be~uili
He re we can fil
I<'lup, Goom, G
tions Barababa
J irnjam, Jumju
Bankwet, Clad
Ndogband, Podl
Anal He, Her,
With this fu
272 o( the book'
languafSe names
Malaya1am and
shared its back
any 10n~er than
In it, the lette r
rlenominate s a1s